AN EXPERIMENTAL STUDY ON NONLINEAR WAVE DYNAMICS: ROUGE WAVE GENERATION AND WAVE BLOCKING

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1 AN EXPERIMENTAL STUDY ON NONLINEAR WAVE DYNAMICS: ROUGE WAVE GENERATION AND WAVE BLOCKING Wen-Yang Hsu, Igor Shugan, Wen-Son Chiang, Ray-Yeng Yang, Hwung-Hweng Hwung 2, Sergey Kuznetsov 3 and Yana Saprykina 3 Wave modulation due to Bengamin Feir Instability (BFI) in the presene of long wave (swell) is investigated in the elaborate experiments. The mehanially generated waves are omposed of three disrete waves; while the newly generated harmonis due to BFI still ombining into disrete spetra with the same frequeny differene step. These stable disrete waves allow us to further examine the basi properties of nonlinear wave-wave interations, suh as the variation of phase speeds and wave numbers along the wave tank together with speed of rests, espeial for the large transient waves. The measured values have a reasonable orrespondene to the available nonlinear model estimations. The tehnique proposed in this study is verified to aurately measure the phase speed and wave number of eah harmoni in the wave group. Phase speeds of short waves have a signifiant amplifiation in the viinity of big transient waves, while the speeds of longer waves are lose to linear values. The relative long wave (storm wave) travelling with short waves (wind waves) dramatially hange the loal kurtosis and skewness and may have an important role for the generation of large transient wave and provide an opportunity for triggering the freak waves. Keywords: Benjamin Feir Instability, giant wave, phase speed, nonlinear wave INTRODUCTION The random sea exhibits nonlinear water wave group behavior. The modulational instability has been onsidered one of the key proesses to determine the nonlinear energy transfers and wave breakings in the deep water (Benjamin and Feir 967). Owing to the diffiulties of haraterizing the modulational instability in the open sea, there is a widespread laboratory experiments whih simplified the real sea state as several disrete waves. Many researhes foused on the evolution of wave modulation and have found some remarkable results, suh as reurrene without downshifting (Fermi- Pasta-Ulam reurrene), temporal and permanent wave energy peak downshifting, aompanied by the wave breaking (Tulin and Waseda 999; Hwung-Hweng Hwung et al. 2), the initial wave steepness value for breaking riteria (Tulin and Waseda, 999; Chiang and Hwung, 27, Hwung et al. 27; Galhenko et al. 2; Banner 24) and wave asade generation by Kartashova & Shugan (2). Furthermore, Benjamin-Feir instability on its further stages up to breaking is intensely studied, been suggested to be one of basi mehanisms of freak wave formation (Kharif and Pelinovsky 23). The prinipal parameter whih influenes the growth rate of sidebands and breaking riteria is the wave steepness. From the temporal and spatial evolution of a group wave it is not straightforward to define the wavenumber at any arbitrary position. As a result, different definitions of wavenumber are proposed to study the wave of interest. For example, Melville (983) applied the Hilbert transform to define the instantaneous frequeny and amplitude for a train of modulated wave evolving in spae and time. His results found that the loal frequenies are highly phase dependent and sometimes go to zero or even negative when the phase reversals ourred. On the other hand, the measurement of time between rests or zero-rossing points assoiated with the wave of interest are ommonly adopted to measure the loal phase speed, wavenumber and wave steepness (Stansell and MaFarlane 22; Grue et al. 23). Many results obtained from the above definitions fous on the harateristis of most energeti wave (e.g., breaking wave), whih is treated as an individual wave and desribed by a single parameter (e.g., phase speed). Nevertheless, the giant wave formation may appear due to the spatial wave fousing (geometrial fousing) and spatio-temporal fousing (dispersion enhanement), where various independent harmoni waves propagating with their own phase speeds (Kharif and Pelinovsky 23). Nonlinear behavior of waves exhibit itself beside the hanging of amplitudes and spetral struture of the wave paket in signifiant variations of kinemati harateristis for every wave omponent suh as phase speed, loal wave number et. There exist some onfusion and gap onerning this issue in the literature, where the wavenumber in the wave group is often studied for individual wave without International Wave Dynamis Researh Center /Tainan Hydraulis Laboratory, National Cheng Kung University, 5 th F., No. 5, Se. 3, Anming Rd., Tainan, 79, Taiwan 2 Department of Hydraulis and Oean Engineering, National Cheng Kung University, No. University Rd., Tainan, 7, Taiwan 3 Institute of Oeanology of Russian Aademy of Siene, Vavilova 38, Mosow, Russia

2 2 COASTAL ENGINEERING 24 onsidering the intrinsi waves. It is interesting to study the nonlinear variation of wave number and phase speed for every harmoni in the wave train along the long feth and to ompare the results with the available linear and nonlinear modeling approahes. Moreover, the quasi-resonant interation between free modes may result in an inrease (in deep water) of kurtosis with respet to its Gaussian value, whih is desribed by a four-wave interation developed by Janssen (23). The modulational instability proess in the oean due to wind waves and long swells has been reognized to be responsible for a fishing boat that was mysteriously sank off the east oast of Japan (Tamura 29). Nevertheless, few laboratory experiments investigated the evolution of large transient waves by onsidering the interation of short and long waves. In the present study, we will generate the nonlinear wave train with long and short waves together and evaluate the ourrene of extreme waves under Bengimin-Feir instability. LABORATORY EXPERIMENTS The experiments were onduted in the wave flume at Tainan Hydraulis Laboratory. The flume is 2 m long, 2 m wide and 2 m deep (see Fig. for the detail setup). A piston-type wavemaker with programmable funtion is equipped at the one end of the wave flume. In the following, we set x= m at the mean position of wave paddle. In order to observe the wave deformation in a uniform water depth, the surfae wave will be measured before the trapezoid bar (x=67~7 m). Twelve apaitane-type wave gauges were used to measure the spatial and temporal evolution of the surfae waves with a sampling rate 5Hz. Starting from the third wave gauge, the wave gauges are plaed with an equal spaing of 4 m. Eah experimental run was reorded for min in real time, inluding the initial rampup. For the present experiments the water depth was fixed at.3 m. Three senarios of group wave were mehanially generated aording to the speified wave form: η( t) = a os(2 π f ) + a os(2 π f ) + a os(2 π f ) () s s Table. Test onditions and the orresponding wave properties f ( Hz ) fs ( Hz ) a / a ω as / a ε = ka kh (a =.7 m) ω / ε Case Case 2.84 NA. NA Case 3.84 NA NA NA NA Figure (a) Shemati view of the experimental setup. EXPERIMENTAL RESULTS Surfae Elevation and Basi Property The initial wave steepness,ε, for Case is.9 and the relative water depth is in the deep water ondition (kh= 3.7). The spatial evolution of the surfae elevation and Fourier amplitudes as a funtion of feth are shown in Fig. 2 and Fig 3. In the initial stage ( <23), the modulational harateristis of the "seeded" sub-harmoni reated a symmetri super-harmoni at f+=.98 Hz. As a result, the initial group wave an be treated as a linear superposition of one arrier wave and two small sidebands together with a low frequeny wave. After, the group wave train exhibits a slow osillation with an inreasing trend that aompanies the symmetri growth of the sidebands until =88. As the wave train further evolves, asymmetri growth of the sideband amplitudes is observed after =22, where some oasional small wave breakings appear in the experiment. This onfirms that the initiation of asymmetri development of sideband amplitudes after initially exponential growth roughly orresponds

3 COASTAL ENGINEERING 24 3 to the onset of wave breaking on a strongly modulated wave train (e.g., Melville 983; Chiang and Hwung 27). Then, the modulation of the wave train inreases until the strongest modulation is reahed at =68, where most of the energy of the wave train is foused into the wave group enter with four energeti individual waves. Moreover, we observe a sequene of large transient waves and the peak wave heights are around.33 m (about.8 times signifiant wave heights under the definition of Hs=4σ). The result reveals a mehanism for the generation of a freak wave resulting from sideband instability. These features will be analyzed in more detail later in Se. 3.3 and 3.4. Around this stage, strong wave breaking is identified on the front of the modulated wave train aording to a video reord of the experiment. The initial wave train in the Case is mehanially generated by three disrete waves, while more than eight disrete waves are observed at the final wave gauge. All the newly generated harmonis have the same seeded frequeny differene (.4 Hz), and they are produed by the quasi resonane nonlinear wave interations suh as BFI and self interations of the higher order Stokes waves. These disrete set of waves in this study allow us to further examine the variation of phase speed of eah harmoni, espeial for the large transient wave. The analysis tehnique will be presented in the next setion. (a) (b) - = = = = = = t (se) f (Hz) Figure 2. (a) The measured surfae elevations at several seleted loations.(b) The orresponding Fourier amplitudes. The dashed line indiates the loal maximum elevation and they are used to alulate the wave speed in Se. 3.4.

4 4 COASTAL ENGINEERING 24.8 f=.56 Hz f=.7 Hz f=.84 Hz f=.98 Hz a i /a Figure 3. The spatial evolution of dimensionless amplitudes Measurement of phase speed The Fast Fourier Transform (FFT) is widely used for water wave problems, espeially in the frequeny spetral evolution of wave groups. In this study, the FFT is used to alulate the phase speed of individual harmoni via two adjaent wave gauges. The traditional approah that traes the time of rest/rossing point from the neighboring wave gauges has been suessfully applied by Lake and Yue (978). Their results showed that the phase speeds of high order bands in the Stokes wave are bounded to the main frequeny wave. Based on their results, though the FFT is a linear proessing method, it still an estimate the nonlinear phase speed within the framework of Stokes wave. It is expeted that the rest-traing approah may fail for the ontinuous spetra wave beause the loal rests in the group wave are usually the superposition of several harmonis. However, for a stationary and disrete frequeny wave spetrum, suh as arrier wave travel with sidebands, the FFT method an effetively separate the group wave into individual harmonis and provide their amplitudes and phases. As a result, the rest traing approah an be applied to the re-onstruted time series of the surfae elevation for single harmoni. The time series of surfae wave at any point an be written as the superposition of several harmonis: N η( xt, ) = a os( ωt + φ ) () i i i i i= where a is amplitude, ω is angular frequeny, k is wavenumber and is phase. In the FFT analysis, the surfae elevation at eah wave gauge is: N η( t) = a os( ωt + φ ) (2) i i ifourier i= where the ϕ ifourier is Fourier phase of eah omponent. The FFT yields the phase relative to the start of the time-domain signal. If we selet the same start time in the FFT analysis for all wave gauges, the ϕ Fourier also represents the quantity of -+ϕ. As a result, the Fourier phase differene between two adjaent wave gauges is mainly ontributed by the k x, in whih the variation of within a short distane is small and is assumed to be zero here. The spae-average wavenumber and the phase speed an be obtained. It is noted that the phase from FFT generally has big onfidene intervals beause the degree of freedom are low. In order to minimize the instability and evaluate the repeatability of signal, we employ the multi-taper method. It separates the signal into several segments and then output their mean value via ensemble average. To verify this method, the Case 3 (monotoni Stokes wave with ε=6)) is seleted as an example to ompare the alulated phase speed from FFT algorithm along with the linear and 3rd Stokes solution. Figure 4 shows the phase speeds obtained from different approahes and all results are normalized by the linear solution. The measured phase speed shows a typial result of Stokes wave that

5 COASTAL ENGINEERING 24 5 is lose to the 3rd Stokes solution. Importantly, the good agreement (< % differene) indiates that the FFT approah an be used to desribe the phase speed evolution of Stokes wave. Finally, the standard deviation is omputed as an indiator of the wave repeatability and unertainty from gauge spaing, frequeny resolution. (see the error bars in Figures 4) Figure 4. The spatial evolution of nondimensional phase speed for Case 3. Solid line: linear solution; dashed line: 3rd Stokes solution; irles: rossing point method; solid points: FFT method Results of group wave Similar to the Case 3, the spatial distribution of phase speed for eah harmoni was alulated, as shown in the Figure 5 and ompare it with the theoretial estimations by the linear, nonlinear 3rd order Stokes solution and multi-resonant model following from the dispersion relations for every harmoni (Phillips 977): σ i = gki ( + Qija jk j ) (3) where Qij are oeffiients of nonlinear Stokes dispersion for the multi-resonant system. The measured phase speed for the low frequeny wave remains stable and is predited well by the linear solution. For the arrier wave, the phase speed generally exhibits an aeleration proess, while the sidebands show several osillations. It is interesting to note that higher frequeny wave has a larger disrepany from the linear solution, whih indiates the nonlinearity (e.g., steepness) is an important fator. Moreover, the faster nonlinear phase speed orresponds to the inreased steepness and they maximally approahes to. times of the linear wave speed (Longuet-Higgins 975). We also note that at ~7 all the phase speeds aelerate until the large transient wave ourred. The inreasing phase speeds are mainly ourred around ~ and ~7, where the two wave breaking events are observed. In general, the measurements are not desribed well either by the 3 rd order Stokes solution or multi-resonant solution. The strong nonlinearity and energy dissipation is onsidered as the main reason and that should be further studied.

6 6 COASTAL ENGINEERING f=.56 Hz.5. f=.7 Hz f=.84 Hz.5. f=.98 Hz Figure 5. The spatial evolution of nondimensional phase speed for Case. Solid line: linear solution; dashed line: 3rd Stokes solution; irles: rossing point method; solid points: FFT method; gray line: the resonant solution Large Transient waves In this setion the generation of large transient waves in the Case is disussed. The arrier wave with sidebands evolves to three foused energy waves due to BFI. In the meanwhile, the low frequeny wave mostly remains its amplitude all along the wave flume and hene it is interesting to note that the four-wave system with equal amplitude is observed around =68. At this position, the time series of the superposition of four deomposed harmonis plotted with the measured data and individual harmoni are also presented, as shown in the Fig. 6. All harmoni waves (the others not shown) ome into the same phase at the maximum elevations. The rest of giant wave is formed by the superposition of Fourier harmonis (marked as dash lines), while the zero elevation is aused by the neutralization of loal rests and troughs (e.g., at t= 7.6 se). Additionally, the wave is traveling inside an envelope with a period Tp, whih is equal to the frequeny differene (e.g., Tp=/ f). The referene Case 2 is a typial BFI senario and the ratio of the maximum wave height to the signifiant wave height is only.4, where there are no giant waves is observed. As a result, the long wave effet is examined by omparing with Case 2, as shown in the Figure 7a. Although the long wave hanges the whole wave struture, however, the growth rate and energy transfers between arrier and sidebands are not greatly enhaned. The long wave inreases the loal steepness and hene hanges the breaking riteria. The maximum disrepanies are observed at the sub-harmoni near the bak-end, where the wave breaking ourred in the Case.The position of the main wave breakings of Case is around 5 m downstream to the Case 2. However, it is interesting to note that the kurtosis of the surfae elevation along the feth is greatly different for Case and Case2. The maximum of the kurtosis takes a plae around x= 6 m and it orresponding to the position of giant wave ourrene, as shown in Fig 7b. The relative long wave (storm wave) travelling with short waves (wind waves) dramatially hange the loal kurtosis and may play an important role for the generation of large transient waves and an provide an opportunity for triggering of the freak waves. A series of giant wave was identified from the last three measurements (traing the maximum surfae elevation starting from =56, 68 to 79, see the dash line in Fig.2). Following the definition of phase speed by traing the rest from adjaent wave gauges, it gives an average group speed of.2 and.9 m/s. Figure 8 shows the 3-D surfae elevation between =56 and 68 in the spae and time

7 COASTAL ENGINEERING 24 7 domain. This plot is generated from Eq. () by taking linear interpolation of the measured amplitudes and wavenumbers and negleting the hange of the initial phase. The maximum surfae elevation at =56 and 68 is marked as number and 4. It is an be found that the giant wave does not ome from point, but grow from a small wave (point 3) and with a phase speed of 2. m/s. By traing the point to 4, it gives the same speed as the result from the Fig.2and whih represents the group speed. The phase speed of giant wave is around two times of group speed and relatively higher than the surrounding waves. Live time of giant wave is relatively short and that is why, probably, there is no intensive breaking in spite of the great steepness. As relative long wave travelling with short waves (wind waves) in the deep water, the experiment antiipates that the giant waves ould appear based on BFI. (a) (b) a /a.5 f=.56 Hz ase ase2 5 5 f=.84 Hz a 2 /a.5 f=.7 Hz 5 5 f=.98 Hz Kurtosis a 3 /a.5 a 4 /a Figure 6. (a) Comparison of the spatial evolution of dimensionless amplitudes for Case and Case 2 (b) and the kurtosis Figure 7. Top panel: Measured data, predited signal from the superposition of four harmonis. Lower panels: time series of eah harmoni from FFT.

8 8 COASTAL ENGINEERING 24 Figure 8. 3-D sketh of spae and time series surfae elevations from th to th Conlusions The nonlinear wave interations within the frame work of BFI are investigated in this study. The disrete wave spetrum allow us to further analyze the kinemati and dynami properties of every omponent in wave group suh as variation of phase speeds and wave numbers. Higher frequeny harmonis exhibit a muh more strong nonlinear properties ompare to long waves behavior. They are generally not well desribed either by the Stokes solution or resonant solutions. The wave breaking generally aompanies by the aeleration of the phase speed of every omponent, however, the modifiation for breaking is not onsidered in the present theories. On the other hand, the multiple nonlinear resonant interation model generally has smaller disrepanies than Stokes solution. The loal kurtosis strongly inreases in the viinity of large transient waves and an serve like a good indiator of extreme wave event. Inluding the energy dissipation in the future would provide the more pratial results. Finally, the relative long wave (storm wave) travelling with short waves (wind waves) in the deep water may play an important role for the generation of large transient wave and provide an opportunity for triggering the freak waves. ACKNOWLEDGMENTS The authors would like to thank the Ministry of Siene and Tehnology of Taiwan (Grant supported by NSC 3-29-I-6-32) and the Aim for the Top University Projet of National Cheng Kung University for their finanial support. REFERENCES Benjamin, T. B. and J. E. Feir 967. 'Disintegration of wave Trains on deep Water.. Theory', Journal of Fluid Mehanis 27: 47 Banner, M. L., Barthelemy X., Fedele F., Allis M., Benetazzo A., Dias F. and Peirson, W. L. 24. 'Linking Redued Breaking Crest Speeds to Unsteady Nonlinear Water Wave Group Behavior', Phys. Rev. Lett., Chiang, W.-S. and H.-H. Hwung 27. 'Steepness effet on modulation instability of the nonlinear wave train.' Physis of Fluids 9(). Grue, J., Clamond, D., Huseby, M., and Jensen, A., 23 'Kinematis of extreme waves in deep water.' Applied Oean Researh 25(6):

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